Modeling Approaches

Modeling Approaches

This page provides quantitative tools for measuring interconnection—because you can’t manage what you can’t measure.

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Delegation Risk Under Correlation

Basic Framework

Define Delegation Risk (DR) as the probability that delegated actions cause unacceptable outcomes.

Independent model:

DR_independent = P(threat) × ∏(1 - effectiveness_i)
                           i

Example: P(threat)=0.1, three 90% layers
DR = 0.1 × 0.1 × 0.1 × 0.1 = 0.0001 (0.01%)

Correlated model (using correlation coefficient ρ):

DR_correlated = P(threat) × P(all layers fail | threat)

Where P(all fail) >> ∏(1 - effectiveness_i) when ρ > 0

The Correlation Tax

Measuring the gap:

Correlation Tax = DR_correlated / DR_independent

Example: If correlation causes joint failure rate of 5% instead of 0.1%
Correlation Tax = 0.005 / 0.0001 = 50×

Your actual risk is 50× higher than independent model suggests.

The Correlation Tax is a single number that communicates how much worse your actual protection is compared to what you’d have with truly independent layers.

Risk Budget Formula

When planning how many layers you need:

Required_layers = log(target_DR) / log(1 - layer_eff × (1 - ρ))

Where:
- target_DR = target delegation risk
- layer_eff = individual layer effectiveness
- ρ = average pairwise correlation

Example:
- Target: 0.001 (99.9% protection)
- Layer effectiveness: 0.9 (90% each)
- Correlation: 0.0 (independent)
- Required layers: 3

With correlation:
- Correlation: 0.5
- Required layers: 7 (not 3!)

With high correlation:
- Correlation: 0.8
- Required layers: 15+ (or impossible)

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Correlation Matrix Approach

Building the Matrix

For N components, maintain an N×N failure correlation matrix:

               Firewall  Ghost   Fingerprint  Tripwire
Firewall         1.0      0.4        0.3        0.2
Ghost            0.4      1.0        0.5        0.3
Fingerprint      0.3      0.5        1.0        0.4
Tripwire         0.2      0.3        0.4        1.0

• Diagonal = 1.0 (component correlated with itself)
• Off-diagonal = correlation in failure modes
• Symmetric for standard correlation (asymmetric if directional)

Estimating Values

Method	Accuracy	Cost	When to Use
Architecture review	Low	Low	Initial estimate
Historical incidents	Medium	Low	If you have incident data
Red team exercises	High	Medium	Regular assessment
Adversarial testing	High	High	Critical systems
Production monitoring	Highest	High	Ongoing measurement

Interpretation Guide

Value	Interpretation	Action
0.0-0.2	Low correlation	Acceptable
0.2-0.4	Moderate	Monitor, consider diversifying
0.4-0.6	High	Investigate root cause
0.6-0.8	Very high	Urgent: redesign required
0.8-1.0	Near-identical	Redundancy is illusory

Visual Representation - Correlation Heatmap

	Firewall	Ghost	Fingerprint	Tripwire
Firewall	1.0	0.5	0.3	0.1
Ghost	0.5	1.0	0.5	0.3
Fingerprint	0.3	0.5	1.0	0.5
Tripwire	0.1	0.3	0.5	1.0

Concern: Ghost-Fingerprint correlation is high (0.5) Action: Investigate shared methodology or infrastructure

A heatmap makes high-correlation pairs immediately visible.

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Dependency Graph with Risk Edges

Building the Graph

Represent interconnections as a weighted directed graph:

flowchart TB
    LLM["LLM Provider<br/>(common cause)"]
    LLM -->|"weight: 0.8"| SF["Semantic Firewall"]
    LLM -->|"weight: 0.8"| GC["Ghost Checker"]
    LLM -->|"weight: 0.8"| VT["Voting Tribunal"]

    SF -->|"data flow: 0.3"| EX[Executor]
    GC -->|"timing: 0.4"| AU[Audit]
    VT -->|"context: 0.5"| RB[Rollback]

    EX --- AU --- RB

    style LLM fill:#ffcccc
    style SF fill:#cce6ff
    style GC fill:#cce6ff
    style VT fill:#cce6ff

Edge Types

Edge Type	Represents	Weight Meaning
Common cause	Shared dependency	Probability both fail if cause fails
Data flow	Information passed	Probability contamination propagates
Timing	Temporal dependency	Probability timing issue cascades
Context	Shared state	Probability context corruption spreads
Resource	Shared pool	Probability contention affects both

Weight Interpretation

Edge weights represent coupling strength:
- 0.0-0.3: Weak coupling (acceptable)
- 0.3-0.6: Moderate coupling (monitor)
- 0.6-1.0: Strong coupling (redesign)

Graph Metrics

Metric	What It Measures	Use
Max path weight	Product of weights on worst path	Worst-case cascade potential
Critical nodes	Nodes with highest betweenness	Single points affecting many components
Clustering coefficient	How interconnected neighborhoods are	Identifies tightly coupled subsystems
Average degree	Connections per node	Overall coupling density

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Swiss Cheese Visualization

The Model

Each defense layer is a slice of cheese with holes (failure modes). Threats pass through when holes align.

flowchart TB
    subgraph Ind["Independent Layers"]
        I1["Layer 1: ● ○ ●"]
        I2["Layer 2: ● ○ ●"]
        I3["Layer 3: ○ ● ●"]
        I1 --> I2 --> I3
        IR["No straight path through"]
    end
    subgraph Cor["Correlated Layers"]
        C1["Layer 1: ● ○ ●"]
        C2["Layer 2: ● ○ ● (same holes!)"]
        C3["Layer 3: ● ○ ● (same holes!)"]
        C1 --> C2 --> C3
        CR["Clear path through!"]
    end

    style Ind fill:#ccffcc
    style Cor fill:#ffcccc
    style IR fill:#99ff99
    style CR fill:#ff9999

• Independent layers: Different holes, no straight path through = threats blocked
• Correlated layers: Same holes everywhere = clear path through = threats pass all layers

(● = solid/blocks threats, ○ = hole/threat passes)

Quantifying Hole Alignment

Hole Alignment Score = (threats passing all layers) / (threats passing any layer)

Independent: Score ≈ 0.001 (holes rarely align)
Correlated:  Score ≈ 0.5   (holes often align)
Maximum:     Score = 1.0   (all holes identical)

How to Measure

1. Generate or collect a set of test threats
2. Run each through each layer independently
3. Count how many pass all layers vs. pass any layer
4. Higher ratio = more correlated holes

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Risk Surface Visualization

The Concept

Plot Delegation Risk as a surface over the space of possible threats:

xychart-beta
    title "Delegation Risk vs Threat Space"
    x-axis "Threat Space" [A, B, C, D, E, F, G, H]
    y-axis "Delegation Risk (log)" 0.001 --> 1.0
    line "Correlated model" [0.1, 0.8, 0.2, 0.9, 0.15, 0.7, 0.2, 0.1]
    line "Independent model" [0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01]

• Gap between curves = correlation tax
• Peaks in correlated model = shared blind spots

Interpretation

• Flat line (independent model): Uniform protection across threat space
• Peaks (correlated model): Areas where all layers fail together
• Gap size: How much the correlation tax varies by threat type
• Peak locations: Identifies specific threat categories to address

Building Risk Surfaces

1. Define axes (e.g., threat complexity × novelty)
2. Sample threats across the space
3. Measure actual failure rate at each point
4. Compare to independent model prediction
5. Visualize as contour plot or 3D surface

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Practical Measurement Protocol

Step 1: Baseline Individual Components

For each component, measure:

• False negative rate (threats missed)
• False positive rate (benign flagged)
• Failure conditions (when it breaks)

Step 2: Test for Pairwise Correlation

For each pair (A, B):

1. Take threats that A misses
2. What fraction does B also miss?
3. Compare to B's baseline miss rate

If B misses more than baseline → positive correlation
If B misses less than baseline → negative correlation (good!)
If B misses same as baseline → independent

Step 3: Test for Higher-Order Effects

Test combinations:
- A + B + C together
- Under load
- With adversarial inputs

Compare to pairwise predictions. Significant deviation indicates higher-order correlation.

Step 4: Calculate Correlation Tax

Correlation Tax = Actual_Joint_Miss_Rate / Predicted_Joint_Miss_Rate

Where:
Predicted = ∏(individual miss rates)
Actual = measured in combined testing

Step 5: Monitor Over Time

Track correlation metrics continuously:

• Joint failure rate
• Correlation coefficients
• Hole alignment score
• Correlation tax

Alert on significant changes.

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Tools and Implementation

Correlation Calculator (Pseudocode)

def calculate_correlation_matrix(components, test_threats):
    n = len(components)
    matrix = np.eye(n)  # Diagonal = 1.0

    for i, comp_a in enumerate(components):
        for j, comp_b in enumerate(components):
            if i >= j:
                continue

            # Get failure sets
            fails_a = {t for t in test_threats if comp_a.fails(t)}
            fails_b = {t for t in test_threats if comp_b.fails(t)}

            # Calculate correlation
            both_fail = len(fails_a & fails_b)
            either_fail = len(fails_a | fails_b)

            if either_fail == 0:
                corr = 0.0
            else:
                # Conditional probability ratio
                corr = both_fail / either_fail

            matrix[i, j] = corr
            matrix[j, i] = corr

    return matrix

def calculate_correlation_tax(components, test_threats):
    # Independent model
    individual_miss_rates = [
        sum(1 for t in test_threats if c.fails(t)) / len(test_threats)
        for c in components
    ]
    predicted_joint = np.prod(individual_miss_rates)

    # Actual joint miss rate
    all_fail = sum(
        1 for t in test_threats
        if all(c.fails(t) for c in components)
    )
    actual_joint = all_fail / len(test_threats)

    if predicted_joint == 0:
        return float('inf') if actual_joint > 0 else 1.0

    return actual_joint / predicted_joint

Dashboard Metrics

For ongoing monitoring, track:

Metric	Target	Alert If
Max pairwise correlation	< 0.3	> 0.5
Average correlation	< 0.2	> 0.3
Correlation tax	< 5×	> 10×
Hole alignment score	< 0.01	> 0.1
Joint failure rate	< individual²	> 10× individual²

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Limitations

What These Models Miss

1. Conditional correlations: Models assume constant correlation; reality is context-dependent
2. Adversarial adaptation: Attackers may specifically target measured blind spots
3. Rare events: Limited test data for extreme cases
4. Model uncertainty: Correlation estimates have confidence intervals
5. Higher-order effects: Most practical methods only capture pairwise

When to Distrust Your Measurements

• Small test sets (< 1000 threats)
• Test distribution doesn’t match production
• Components have changed since measurement
• No adversarial testing included
• Measurement during non-representative conditions

Appropriate Conservatism

When in doubt:
- Assume correlation is higher than measured
- Assume measured correlation has wide confidence interval
- Use stress-test correlation, not normal-operation correlation
- Add safety margin to correlation tax estimates

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See also:

• Types of Correlation - What types of correlation exist
• Challenge Categories - Sources to measure
• Worked Examples - Applying these methods